What is the angle between two intersecting lines?

The angle between two intersecting lines is the smallest angle formed between them at the point of intersection. It can be calculated using the slopes of the lines if they are given in the slope-intercept form.

Can the angle between two lines be negative?

No, the angle between two lines is always a positive value. The formula for tan θ gives the absolute value, ensuring the angle is positive.

What happens if one of the lines is vertical?

If one of the lines is vertical, its slope is undefined. The angle between a vertical line and any other line can be calculated by considering the vertical line's slope as approaching infinity and using geometric reasoning to find the angle.

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Parabola

The parabola, a conic section, is a symmetric U-shaped curve. It appears in everyday scenarios and physics, showcasing its versatility.

Angle Between Two Lines

When two straight lines intersect, they create two sets of angles: one pair of acute angles and one pair of obtuse angles. The exact values of these angles depend on the slopes of the intersecting lines.

It is important to note that if one of the lines is parallel to the y-axis, the angle between the lines cannot be calculated using the slope formula, as the slope of a line parallel to the y-axis is undefined.

1.0Formulas for Angle Between Two Lines

If θ is the angle between two intersecting lines given by the equations y₁ = m₁x + c₁ and y₂ = m₂x + c₂, the angle θ can be calculated using the slopes m₁and m₂of these lines. The formula to find θ is:

$tan θ = \pm \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}$

2.0Derivation of the Formula

Consider two non-vertical lines, L₁ and L₂ with slopes m₁ and m₂ respectively, and

$α_{1}$ and $α_{2}$ be the angle made by the line L₁ and L₂ respectively.

$∴ tan α_{1} = m_{1}$ and tan α₂ = m₂

Let θ be the angle between the lines L₁ and L₂ and φ, adjacent angle to θ.

Derivation of formula for the angle between 2 lines

From the figure, $α_{2} = α_{1} + θ$

$∴ θ = α_{2} - α_{1} an d α_{1}, α_{2} \neq = 9 0^{\circ}$

$∴ tan θ = tan (α_{2} - α_{1})$

$\frac{t a n α _{2} - t a n α _{1}}{1 + t a n α _{2} t a n α _{1}} = \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}$

And $φ = 18 0^{\circ} - θ = π - θ$

$∴ tan φ = tan (180 - θ)$

$= - tan θ = - (\frac{m _{2} - m _{1}}{1 + m _{1} m _{2}})$

∴ If θ is acute, tan θ is +ve and if θ is obtuse, tan θ is –ve.

∴ If θ is the acute angle between the given line, then

$tan θ = \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}_{1 + m_{1} m_{2} \neq = 0}$

Hence, angle between the line $tan θ = \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}$

Related Video:

3.0Some Important Points Related to Angle Between Two Lines

Acute angle between two lines having gradients m₁ and m₂ is $tan^{- 1} \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}$ , therefore obtuse angle is $π - tan^{- 1} \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}} .$
Lines are parallel when $m_{1} = m_{2} .$
Lines are perpendicular when $m_{1} m_{2} = - 1.$
Acute angle of a line with x-axis = $tan^{- 1} ∣ m ∣.$
Acute angle of a line with y-axis = $\frac{π}{2} - tan^{- 1} ∣ m ∣ = cot^{- 1} ∣ m ∣ = tan^{- 1} \frac{1}{∣ m ∣} .$

4.0Angle Between Two Lines in Three-Dimensional Space

If $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ are the two lines then angle between two lines in a three dimensional Space can be represented as:-

$cos θ = \frac{a _{1} a _{2} + b _{1} b _{2} + c _{1} c _{2}}{a _{1}^{2} + b _{1}^{2} + c _{1}^{2} a _{2}^{2} + b _{2}^{2} + c _{2}^{2}}$

5.0Solved Examples of Angle Between Two Lines

Example 1: Find the angle between the lines given by y = 2x + 3 and $y = - \frac{1}{2} x + 1$ .

Solution:

From the given equation

y = 2x + 3, slope $m_{1} = 2$

y= $- \frac{1}{2} x + 1$ , slope $m_{2} = - \frac{1}{2}$

By using the formula of angle between two lines.

$tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}$

$\Rightarrow tan θ = \frac{2 - ( - \frac{1}{2} )}{1 + 2 \cdot ( - \frac{1}{2} )}$

$\Rightarrow tan θ = \frac{2 + \frac{1}{2}}{1 - 1}$

$\Rightarrow tan θ = \frac{\frac{5}{2}}{0}$

$\Rightarrow tan θ$ = not defined

So, $θ = 9 0^{\circ}$

Example 2: Find the angle between the lines given by y = 3x + 4 and y = –2x + 1.

Solution:

From the given equation

y = 3x + 4, slope m₁ = 3

y = –2x + 1, slope m₂ = –2

Using formula

$tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}$

$\Rightarrow tan θ = \frac{3 - ( - 2 )}{1 + 3 \cdot ( - 2 )}$

$\Rightarrow tan θ = \frac{3 + 2}{1 - 6}$

$\Rightarrow tan θ = \frac{5}{- 5}$

$\Rightarrow tan θ = 1$

$∴ θ = tan^{- 1} (1) = 4 5^{\circ}$

Example 3: Find the angle between the lines given by y = 2x + 3 and y = 4.

Solution:

From the given equations

y = 2x + 3 slope m₁ = 2

y = 4 slope m₂ = 0 {which is a horizontal line}

Using formula

$tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}$

$\Rightarrow tan θ = \frac{2 - 0}{1 + 2 ( 0 )} \Rightarrow tan θ = \frac{2}{1}$

$\Rightarrow θ = tan^{- 1} (2)$

Example 4: Find the angle between the lines given by y= $\frac{- 3}{2} x + 5$ and $y = \frac{2}{3} x$ -1.

Solution:

From the given equation

$y = \frac{- 3}{2} x + 5$ , slope $m_{1} = \frac{- 3}{2}$

$y = \frac{2}{3} x - 1$ , slope $m_{2} = \frac{2}{3}$

Using formula

$tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}$

$\Rightarrow tan θ = \frac{- \frac{3}{2} - \frac{2}{3}}{1 + ( - \frac{3}{2} ) ( \frac{2}{3} )}$

$\Rightarrow tan θ = \frac{\frac{- 13}{6}}{1 - 1}$

$\Rightarrow tan θ = \frac{\frac{- 13}{6}}{0}$

$\Rightarrow tan θ$ = not defined.

So, $θ = 9 0^{\circ}$

Example 5: Find the acute angle between the lines given by y = 3x + 2 and $y = - \frac{1}{2} x + 1$ .

Solution:

From the given equation

y=3 x+2 $\Rightarrow$ slope $m_{1} = 3$

$y = - \frac{1}{2} x + 1 \Rightarrow$ slope $m_{2} = - \frac{1}{2}$

Using formula

$tan θ = \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}$

$\Rightarrow tan θ = \frac{- \frac{1}{2} - ( 3 )}{1 + 3 ( - \frac{1}{2} )}$

$\Rightarrow tan θ = \frac{- \frac{7}{2}}{1 - \frac{3}{2}}$

$\Rightarrow tan θ = \frac{- \frac{7}{2}}{- \frac{1}{2}}$

$\Rightarrow tan θ = 7$

$\Rightarrow θ = tan^{- 1} (7)$

Since tan θ is +ve so θ is acute. So the acute angle between the lines is $tan^{- 1} (7)$ .

6.0Practice problems of Angle Between Two Lines

1. Find the angle between the lines given by y = –3x + 2 and $y = - \frac{1}{3} x - 4$

2. Find the angle between the lines given by y = 4x + 7 and y = x – 2.

3. Find the angle between the lines y = 2x + 3 and y= $- \frac{1}{2} x - 4.$

4. Calculate the angle between the lines given by y = –x + 7 and y = 2x – 1.

5. Find the acute angle between the lines given by 4x + 3y = 7 and 3x – 4y = 8.

7.0Sample Questions on Angle Between Two Lines

Q1. What is the angle between two perpendicular lines?

Ans: The angle between two perpendicular lines is $9 0^{\circ}$ . For two lines to be perpendicular, the product of their slopes must be –1.

Q2. How do you find the angle between two lines in a plane?

Ans: To find the angle θ between two lines with slopes $m_{1}$ and $m_{2}$ , use the formula:

$tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}$ . Then, find θ by taking the arctan of the result.

Also Read:

Angle Bisector Theorem	Statistics	Pascal’s triangle
Distance Between Two Points	Tangent and Normal	3D Geometry
Angle Between Two Lines	Circle	Section Formula

Frequently Asked Questions

Join ALLEN!

(Session 2026 - 27)

Name

Mobile Number

Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Double Integral

Double integration is a mathematical process used to compute the integral of a function over a two-dimensional region.

Quadratic Equations

the quadratic equation is defined as a polynomial equation of the second degree. This implies that they consist...

Altitude of a Triangle

The altitude of a triangle is a perpendicular segment drawn from a vertex to the line containing the opposite side (base). It represents the height of the triangle relative to that base.

Matrices

A Matrix is an organized rectangular array comprising numbers, symbols, or expressions that are systematically arranged in rows and columns.

Fibonacci Numbers

Fibonacci numbers are a sequence of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1.

Fibonacci Sequence

The Fibonacci sequence is a sequence of numbers in which each number (called a Fibonacci number) is the sum of the two preceding ones.

Dot Product of Two Vectors

The dot product, often referred to as the scalar product, is a foundational concept in vector algebra.

Parabola

The parabola, a conic section, is a symmetric U-shaped curve. It appears in everyday scenarios and physics, showcasing its versatility.

Angle Between Two Lines

When two straight lines intersect, they create two sets of angles: one pair of acute angles and one pair of obtuse angles. The exact values of these angles depend on the slopes of the intersecting lines.

It is important to note that if one of the lines is parallel to the y-axis, the angle between the lines cannot be calculated using the slope formula, as the slope of a line parallel to the y-axis is undefined.

1.0Formulas for Angle Between Two Lines

If θ is the angle between two intersecting lines given by the equations y₁ = m₁x + c₁ and y₂ = m₂x + c₂, the angle θ can be calculated using the slopes m₁and m₂of these lines. The formula to find θ is:

$tan θ = \pm \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}$

2.0Derivation of the Formula

Consider two non-vertical lines, L₁ and L₂ with slopes m₁ and m₂ respectively, and

$α_{1}$ and $α_{2}$ be the angle made by the line L₁ and L₂ respectively.

$∴ tan α_{1} = m_{1}$ and tan α₂ = m₂

Let θ be the angle between the lines L₁ and L₂ and φ, adjacent angle to θ.

From the figure, $α_{2} = α_{1} + θ$

$∴ θ = α_{2} - α_{1} an d α_{1}, α_{2} \neq = 9 0^{\circ}$

$∴ tan θ = tan (α_{2} - α_{1})$

$\frac{t a n α _{2} - t a n α _{1}}{1 + t a n α _{2} t a n α _{1}} = \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}$

And $φ = 18 0^{\circ} - θ = π - θ$

$∴ tan φ = tan (180 - θ)$

$= - tan θ = - (\frac{m _{2} - m _{1}}{1 + m _{1} m _{2}})$

∴ If θ is acute, tan θ is +ve and if θ is obtuse, tan θ is –ve.

∴ If θ is the acute angle between the given line, then

$tan θ = \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}_{1 + m_{1} m_{2} \neq = 0}$

Hence, angle between the line $tan θ = \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}$

Related Video:

3.0Some Important Points Related to Angle Between Two Lines

Acute angle between two lines having gradients m₁ and m₂ is $tan^{- 1} \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}$ , therefore obtuse angle is $π - tan^{- 1} \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}} .$
Lines are parallel when $m_{1} = m_{2} .$
Lines are perpendicular when $m_{1} m_{2} = - 1.$
Acute angle of a line with x-axis = $tan^{- 1} ∣ m ∣.$
Acute angle of a line with y-axis = $\frac{π}{2} - tan^{- 1} ∣ m ∣ = cot^{- 1} ∣ m ∣ = tan^{- 1} \frac{1}{∣ m ∣} .$

4.0Angle Between Two Lines in Three-Dimensional Space

If $a_{1} x + b_{1} y + c_{1} = 0$ and $a_{2} x + b_{2} y + c_{2} = 0$ are the two lines then angle between two lines in a three dimensional Space can be represented as:-

$cos θ = \frac{a _{1} a _{2} + b _{1} b _{2} + c _{1} c _{2}}{a _{1}^{2} + b _{1}^{2} + c _{1}^{2} a _{2}^{2} + b _{2}^{2} + c _{2}^{2}}$

5.0Solved Examples of Angle Between Two Lines

Example 1: Find the angle between the lines given by y = 2x + 3 and $y = - \frac{1}{2} x + 1$ .

Solution:

From the given equation

y = 2x + 3, slope $m_{1} = 2$

y= $- \frac{1}{2} x + 1$ , slope $m_{2} = - \frac{1}{2}$

By using the formula of angle between two lines.

$tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}$

$\Rightarrow tan θ = \frac{2 - ( - \frac{1}{2} )}{1 + 2 \cdot ( - \frac{1}{2} )}$

$\Rightarrow tan θ = \frac{2 + \frac{1}{2}}{1 - 1}$

$\Rightarrow tan θ = \frac{\frac{5}{2}}{0}$

$\Rightarrow tan θ$ = not defined

So, $θ = 9 0^{\circ}$

Example 2: Find the angle between the lines given by y = 3x + 4 and y = –2x + 1.

Solution:

From the given equation

y = 3x + 4, slope m₁ = 3

y = –2x + 1, slope m₂ = –2

Using formula

$tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}$

$\Rightarrow tan θ = \frac{3 - ( - 2 )}{1 + 3 \cdot ( - 2 )}$

$\Rightarrow tan θ = \frac{3 + 2}{1 - 6}$

$\Rightarrow tan θ = \frac{5}{- 5}$

$\Rightarrow tan θ = 1$

$∴ θ = tan^{- 1} (1) = 4 5^{\circ}$

Example 3: Find the angle between the lines given by y = 2x + 3 and y = 4.

Solution:

From the given equations

y = 2x + 3 slope m₁ = 2

y = 4 slope m₂ = 0 {which is a horizontal line}

Using formula

$tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}$

$\Rightarrow tan θ = \frac{2 - 0}{1 + 2 ( 0 )} \Rightarrow tan θ = \frac{2}{1}$

$\Rightarrow θ = tan^{- 1} (2)$

Example 4: Find the angle between the lines given by y= $\frac{- 3}{2} x + 5$ and $y = \frac{2}{3} x$ -1.

Solution:

From the given equation

$y = \frac{- 3}{2} x + 5$ , slope $m_{1} = \frac{- 3}{2}$

$y = \frac{2}{3} x - 1$ , slope $m_{2} = \frac{2}{3}$

Using formula

$tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}$

$\Rightarrow tan θ = \frac{- \frac{3}{2} - \frac{2}{3}}{1 + ( - \frac{3}{2} ) ( \frac{2}{3} )}$

$\Rightarrow tan θ = \frac{\frac{- 13}{6}}{1 - 1}$

$\Rightarrow tan θ = \frac{\frac{- 13}{6}}{0}$

$\Rightarrow tan θ$ = not defined.

So, $θ = 9 0^{\circ}$

Example 5: Find the acute angle between the lines given by y = 3x + 2 and $y = - \frac{1}{2} x + 1$ .

Solution:

From the given equation

y=3 x+2 $\Rightarrow$ slope $m_{1} = 3$

$y = - \frac{1}{2} x + 1 \Rightarrow$ slope $m_{2} = - \frac{1}{2}$

Using formula

$tan θ = \frac{m _{2} - m _{1}}{1 + m _{1} m _{2}}$

$\Rightarrow tan θ = \frac{- \frac{1}{2} - ( 3 )}{1 + 3 ( - \frac{1}{2} )}$

$\Rightarrow tan θ = \frac{- \frac{7}{2}}{1 - \frac{3}{2}}$

$\Rightarrow tan θ = \frac{- \frac{7}{2}}{- \frac{1}{2}}$

$\Rightarrow tan θ = 7$

$\Rightarrow θ = tan^{- 1} (7)$

Since tan θ is +ve so θ is acute. So the acute angle between the lines is $tan^{- 1} (7)$ .

6.0Practice problems of Angle Between Two Lines

1. Find the angle between the lines given by y = –3x + 2 and $y = - \frac{1}{3} x - 4$

2. Find the angle between the lines given by y = 4x + 7 and y = x – 2.

3. Find the angle between the lines y = 2x + 3 and y= $- \frac{1}{2} x - 4.$

4. Calculate the angle between the lines given by y = –x + 7 and y = 2x – 1.

5. Find the acute angle between the lines given by 4x + 3y = 7 and 3x – 4y = 8.

7.0Sample Questions on Angle Between Two Lines

Q1. What is the angle between two perpendicular lines?

Ans: The angle between two perpendicular lines is $9 0^{\circ}$ . For two lines to be perpendicular, the product of their slopes must be –1.

Q2. How do you find the angle between two lines in a plane?

Ans: To find the angle θ between two lines with slopes $m_{1}$ and $m_{2}$ , use the formula:

$tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}$ . Then, find θ by taking the arctan of the result.

Also Read:

Angle Bisector Theorem	Statistics	Pascal’s triangle
Distance Between Two Points	Tangent and Normal	3D Geometry
Angle Between Two Lines	Circle	Section Formula

Frequently Asked Questions

What is the angle between two intersecting lines?

Can the angle between two lines be negative?

What happens if one of the lines is vertical?

Join ALLEN!

Related Articles:-

Double Integral

Quadratic Equations

Altitude of a Triangle

Matrices

Fibonacci Numbers

Fibonacci Sequence

Dot Product of Two Vectors

Parabola

Angle Between Two Lines

1.0Formulas for Angle Between Two Lines

2.0Derivation of the Formula

3.0Some Important Points Related to Angle Between Two Lines

4.0Angle Between Two Lines in Three-Dimensional Space

5.0Solved Examples of Angle Between Two Lines

6.0Practice problems of Angle Between Two Lines

7.0Sample Questions on Angle Between Two Lines

Table of Contents

Frequently Asked Questions

What is the angle between two intersecting lines?

Can the angle between two lines be negative?

What happens if one of the lines is vertical?

Join ALLEN!

Related Articles:-

Double Integral

Quadratic Equations

Altitude of a Triangle

Matrices

Fibonacci Numbers

Fibonacci Sequence

Dot Product of Two Vectors

Parabola

Angle Between Two Lines

1.0Formulas for Angle Between Two Lines

2.0Derivation of the Formula

3.0Some Important Points Related to Angle Between Two Lines

4.0Angle Between Two Lines in Three-Dimensional Space

5.0Solved Examples of Angle Between Two Lines

6.0Practice problems of Angle Between Two Lines

7.0Sample Questions on Angle Between Two Lines

Table of Contents